Ultrafast imaging of polariton propagation and interactions

Semiconductor excitations can hybridize with cavity photons to form exciton-polaritons (EPs) with remarkable properties, including light-like energy flow combined with matter-like interactions. To fully harness these properties, EPs must retain ballistic, coherent transport despite matter-mediated interactions with lattice phonons. Here we develop a nonlinear momentum-resolved optical approach that directly images EPs in real space on femtosecond scales in a range of polaritonic architectures. We focus our analysis on EP propagation in layered halide perovskite microcavities. We reveal that EP–phonon interactions lead to a large renormalization of EP velocities at high excitonic fractions at room temperature. Despite these strong EP–phonon interactions, ballistic transport is maintained for up to half-exciton EPs, in agreement with quantum simulations of dynamic disorder shielding through light-matter hybridization. Above 50% excitonic character, rapid decoherence leads to diffusive transport. Our work provides a general framework to precisely balance EP coherence, velocity, and nonlinear interactions.

mL N, N-dimethylformamide (Thermo Scientific™, anhydrous, 99.8%). Subsequently, (BA)2MAPb2I7 slabs are grown between two DBR substrates by slow solvent evaporation in an Argon-protected atmospheres at 60 °C for an hour. WSe2 sample preparation. The self-hybridized WSe2 cavity sample was prepared by direct Scotch-tape mechanical exfoliation of bulk high-purity flux-grown WSe2 crystals 3,4 onto a borosilicate cover glass. Flakes of the right thickness were found using atomic force microscopy (Bruker Dimension FastScan Atomic Force Microscope).

6,13-Bis(triisopropylsilylethynyl)pentacene (TIPS-pentacene) plexciton sample preparation.
A 30 nm silver film was deposited on cover glass by e-beam deposition (at 0.05 nm/s), creating a plasmonic film. The silver is then coated with a 3 nm Al2O3 top layer that acts to both protect the silver and prevent direct charge-transfer interactions with molecules. Al2O3 that was grown by atomic layer deposition (SAVANNAH 200, Cambridge Nano Tech Inc.). TIPS-Pentacene (≥ 99%, 20 mg) and poly(methyl methacrylate) (PMMA, M.W. 35000, 6 mg) was dissolved in toluene (1 mL) under constant stirring. The TIPS-pentacene/PMMA solution is then spin-coated on the silvercoated cover glass at 3000 rpm for 60 seconds.
MUPI is shown in Figure S1. A 40 W Yb:KGW ultrafast regenerative amplifier (Light Conversion Carbide, 40W, 1030 nm fundamental, 1 MHz repetition rate) seeds an optical parametric amplifier (OPA, Light Conversion, Orpheus-F) with a signal tuning range of 640 -940 nm and an average pulsewidth of 60 fs. For non-resonant excitation experiments, the second harmonic of the fundamental (515 nm) is used as a pump pulse, and the OPA signal is used as probe. For resonant excitation (single-color) experiments, the OPA signal is split in pump and probe beams using a beamsplitter. Dispersion of the OPA signal caused by refractive optics (in particular the microscope objective) is partially pre-compensated using a pair of chirped mirrors (Venteon DCM7). For all experimental configurations, the pump pulse train is modulated at 647 Hz using an optical chopper and is sent collimated into a high numerical-aperture objective (Leica HC Plan Apo 63x, 1.4 NA oil immersion), resulting in diffraction-limited excitation on the sample. Typical pump fluence incident on the semiconductor is 5 μJ/cm 2 . The probe is sent to a computercontrolled mechanical delay line for control over pump-probe time delay, and is combined with the pump beam through a dichroic mirror. An f = 250 mm widefield lens is inserted prior to the dichroic mirror to focus the probe in the back focal plane of the objective, resulting in widefield illumination of the sample. A tilting mirror placed one focal length upstream of the widefield lens allows tuning the angle at which the widefield probe illuminates the sample, thus allowing probing at any momentum up to a maximum of k/k0 = 1.4, limited by the numerical aperture of the objective.
A beamsplitter collects the backscattered light from the sample through the objective, directing the light to two different detection paths. For angle-resolved linear and transient reflectance ( Figure 1 h-j of the main text), the back focal plane of the objective is projected on the entrance slit of a home-built prism spectrometer using a pair of lenses (f1 = 300 mm and f2 = 100 mm), as depicted in Figure S1. For real-space MUPI imaging, this projected back focal plane image is Fourier transformed again into real-space using a 150 mm lens, forming an image on a CMOS camera (Blackfly S USB3, BFS-U3-28S5M-C). Both the spectrometer camera and the real-space camera are triggered at double the pump modulation rate, allowing the consecutive acquisition of images with the pump ON followed by the pump OFF. Consecutive frames are then processed according to (pump on/pump off -1).
For cryogenic measurements, the same light source is used, but the sample is placed in a closeloop Montana Instruments s100 cryostation equipped with a cryo-optic objective (in-vacuum but room-temperature objective). The objective is a Zeiss LD EC Epiplan-Neofluar 100x/0.90 DIC M27 (NA = 0.9).
The configuration used for ultrafast angle-resolved transient reflectance experiments (Figure 1h,i) is almost identical to that described above, except that the probe is a supercontinuum white light generated by focusing the fundamental 1030 nm beam from the regenerative amplifier in a 5.0 mm yttrium aluminum garnet flat window (YAG, undoped, orientation[111], EKSMA Optics). For this experiment, the probe is sent collimated into the objective and overfills the back aperture of the objective. This configuration ensures that all angles allowed by the numerical aperture of the objective are collected, allowing the data in Figures 1h,i to be collected in a single pump/probe image pair, i.e. without having to scan across angles. The inset shows the early pump-probe time delay signal. The estimated instrument response function from the fit is 129 fs. Figure S3. Sample optical images. Images for a 0.667 μm (a) and 0.696 μm (b) thick BA2(MA)Pb2I7 crystal flakes enclosed in a Au-Ag cavity, corresponding to sample 1 and sample 2 in main text. This image was collected in reflection using a halogen white light lamp and a 40X objective. The scale bar is 50 μm.

Coupled oscillator model, scattering matrix simulations, and polariton lifetimes
As has been shown in the recent literature [5][6][7][8][9] , multi-mode cavities comprising a semiconductor slab of finite thickness (i.e. not a monolayer) in the strong-coupling regime are best fit using a coupled oscillator model described by a 2N-dimension block-diagonal Hamiltonian for N cavity modes. For the three modes observed in the dispersion in our case, we thus model the dispersion using the following coupled oscillator model: Where Uex is the exciton energy (2.14 eV), Hn is the energy for each cavity mode, is the interaction energy between each cavity mode and the exciton, and Epol is the polariton energy. The implication of this model is that for this sample geometry, the cavity modes don't interact with each other. As a result, each polariton branch can be modeled as a 2x2 coupled oscillator between each cavity mode and the exciton. Figure S4a (dashed lines) displays the result of such a coupled oscillator fit overlaid on the experimental dispersion for the three lower polariton branches. The fit provides a value of = 137.5 meV, i.e., a Rabi splitting of 275 meV. The dispersion shows Slike bending (flattening) of the lower polariton branches toward higher momenta, avoiding crossing of the exciton linea characteristic signature of strong light-matter interaction. Figure S4b shows the result of a scattering matrix method (SMM) calculation, performed using the open-source S4 package, 10 using the perovskite slab thickness and the known dielectric functions of the metallic mirrors and BA2(MA)Pb2I7. The experimental dispersion, coupled oscillator model and SMM calculations are all in excellent agreement with one another. Finally, Figure S4c shows the full decomposition of the coupled oscillator model for each mode. Note that upper polariton branches are not visible in the experimental dispersion because the material is highly absorptive above bandgap, as commonly observed in the perovskite polariton literature 11,12 , and as also captured in the SMM calculations (Fig. S4b). The exciton content of the polariton for each mode is obtained by the eigenstate normalization: Where A and X are the Hopfield coefficients associated with the relative photon and exciton content of the polariton.
The EP lifetimes for the various polaritons probed in Figure 2 of the main text are estimated using two methods and tabulated in Table S1. We note that EP lifetimes are difficult to calculate exactly due to various losses and disorder not accounted for in simple models; however, the EP lifetime is not used as a parameter in any of our analyses. First, we can estimate the EP lifetime using: is the exciton lifetime (~6 ns in our case, figure S5), is the cavity lifetime, L is the cavity thickness, is the angle relative to the normal of the interface, n is the refractive index of the semiconductor, and ( ) are the Fresnel reflection coefficients for the top and bottom interfaces of the cavity.
The second method we use is to calculate the lifetime through the inverse Lorentzian linewidth (Γ) of the experimental dispersion, = ℏ/Γ which provides a lower bound for the polariton lifetime. Most lifetimes range from ~100 -300 fs. For completeness, we also calculate the inverse linewidth obtained from SMM calculations, which provides an upper lifetime limit assuming no disorder and losses not accounted for in the dielectric functions of the system.  Finally, the expected group velocity g of the polariton is calculated by first-order differentiation of the dispersion relation: Where is the resonance (angular) frequency of the polariton and k is the momentum.

Supplementary Note 1 Exciton and EP transport analysis and spatial precision
The mean squared displacement (msd) of the photoexcited species in spatiotemporal microscopies is defined as: [13][14][15] where σ is the Gaussian width, t is the pump-probe time delay, D is the diffusivity, and is an exponent characterizing the transport regime. For diffusive transport, α = 1; for sub-diffusive transport, α < 1; in the limit of ballistic transport, α = 2 (corresponding to distance ∝ time). 14,15 Thus, the msd enables us to unambiguously characterize different transport regimes observed for excitons and polaritons. The spatial precision with which exciton or polariton transport can be characterized is not defined by the diffraction limit, but rather by the signal-to-noise ratio of the measurement 14 . For example, we can define a propagation length L = √msd which has an error This error is the spatial precision of our measurement, and is set by the precision with which the variance can be fit. Our measurements typically yield sub-30 nm spatial precision. Figure S6. Bare exciton transport in BA2(MA)Pb2I7 taken with a probe at the exciton resonance at k = 0. The population profiles were obtained through radial averaging of the data shown in Figure 1c of the main text. Gaussian fits to the population profiles allow extracting σ and thus the msd. In our 2D halide perovskites, exciton transport is observed to be sub-diffusive, and is fit to a trap-limited, exponentially-decaying diffusivity in agreement with recent reports 16 , as detailed in the main text. The pump fluence for this dataset is 4.44 μJ/cm 2 .
For polaritons probed at finite k, the propagation is one-sided; the variance 2 ( ) can either be obtained by a single-sided Gaussian fit to the polariton profile, or by fitting the arrival time of the polariton at a specific distance away from the excitation spot, as shown in Figure S7. Both approaches give almost identical results in most datasets, but we have found the method shown in Figure S7 to be more robust in measurements with low signal-to-noise ratio. Since the rise time at each location is a convolution of the instrument response function with the delayed arrival of the EP, the rise times are fit using a Gaussian function convolved with a bi-exponential decay (the second decay is a nanosecond decay component to account for the offset), where A1, A2, 1 , 2 are fit amplitudes and decay times, w is the instrument response function width, and t0 is defined as the EP arrival time. Additional fits and datasets are provided in Figure S15-17. Note that this fitting procedure provides almost identical results to extracting the EP arrival time as the time at which the EP signal rises to half its maximum amplitude (schematically illustrated with the black line in panel b). (c) Resulting distance vs delay plot extracted from the fitting procedure, showing linear behavior. Note that the msd (rather than distance) is plotted in the main text. Error bars are one standard deviation

Monte-Carlo simulations of polariton transport
To model the imaging patterns observed in MUPI, Monte-Carlo simulations such as that shown in Figure 1e of the main text were carried out. In these simulations, imaged EP 'particles' are assumed to move ballistically at the group velocity extracted from the polariton dispersion, and along the wavevector probed by the probe field, until they are elastically scattered by the lattice. For nonresonant excitation, the EP injection rate is modeled according to the empirical rise-time of the signal; this finite rise-time reflects population scattering from the long-lived exciton reservoir to the lower polariton branch ( Figure S8). For resonant excitation, the EP injection rate is modeled after the instrument response function ( Figure S2). In the Monte-Carlo model, the EP particles are initialized having the same group velocity, and with a spatial Gaussian distribution where r is the distance of the EPs to the excitation origin and is the standard deviation of the distribution, which is set as the diffraction limit for the pump wavelength used. The directions of EPs are randomized in the initialization. At each time step of dt = 10 fs, we calculate the probability of EP-lattice scattering and EP loss with exp (− / phonon ) and exp (− / ), where phonon and are the EP-lattice scattering time and polariton lifetime, respectively. In these simulations, phonon is empirically tuned to match the transport behavior observed in our experiments. The elastically scattered EPs are then assigned a new random angle ( Figure S9a). To reproduce MUPI images, wherein the probe selects only EPs with specific wavevectors, we selectively plot EPs populated within a finite range of angles, even though the simulation includes all EPs with different momenta. This angle range is extracted from the k-width of the EP dispersion curve in the angle-resolved reflectance spectra, fitted by a Lorentzian function. All simulations are averaged over 120 independent runs.
For EP simulations of resonant excitation (Figure 3f), where the EP population can saturate within the pump pulse width, we additionally introduce EP-EP scattering. We estimate the scattering time using the semiclassical approximation 17 : where σ is the EP scattering cross-section, n is the polariton density, and v is the group velocity. σ was calculated by = EX | | 4 , where EX is the exciton-exciton scattering cross-section (4 nm, assigned as the exciton Bohr radius 18 ). Figure S8. The temporal rise-time for EPs probed at different energies (corresponding to the data for sample 1 in Figure 2) following above-gap excitation.
To model the MUPI image formation process, the spatial EP population distribution is convolved with a filter function that models the interference between the widefield probe (plane wave) and spherical wave scattering from the EP population ( Figure S9), i.e.: Where I is the light intensity detected on the CMOS camera, A and C are the wave amplitudes, k, ω are respectively the spatial angular frequency and circular frequency of the electromagnetic wave. is the wavevector, is the position vector, and represents the angle between the vectors. is an arbitrary phase tuned to match the experimental profile. Subscripts p and sp correspond to plane wave and spherical wave, respectively. The resulting images (shown in Figure 1e of the main text) for different time delays closely reproduce the characteristic MUPI profile of interferencelike features near the pump excitation location, as well as a ballistically expanding wavefront along the probed wavevector.
All Monte-Carlo simulations were carried out in python. To simulate and analyze the angle-resolved reflectance profiles presented in Figures 1b, h, i of the main text, we turn to a scattering matrix approach, which we perform using the open-source S4 package. 10 We first start by simulating the ground state (linear) angle-resolved reflectance profile. Input simulation parameters are the ground state dielectric function of BA2(MA)Pb2I7 (figure S10a) which we obtain from the literature; 5 incident light angle (varied from 0 to 90 degrees) and polarization (s-polarized); the cavity thickness (0.667 μm); and mirror layers (30 nm gold and 150 nm silver), with light incident through the gold. Figure S10c shows the resulting linear angleresolved reflectance profile, which agrees well with the experimental data ( Figure S10b).
The experimental transient angle-resolved reflectance profile ( Figure 1h of the main text) displays a ~10 meV blueshift of all ground state polariton branches almost uniformly across all momenta. We hypothesize that this uniform blueshift arises due to a well-known pump-induced shift of the dielectric function in 2D halide perovskites. This pump-induced shift, typically on the order of 5-15 meV, 20 occurs due to excitonic many-body interactions.
To test our hypothesis, we used the scattering matrix approach to calculate the angle-resolved reflectance profile using the excited-state dielectric function of BA2(MA)Pb2I7. Considering that not all excitons are populated, the excited state dielectric function is generated by averaging 3% of a 10 meV blueshifted dielectric with the ground state dielectric function. We then simulate the transient (differential) angle-resolved reflectance profile by calculating Where Rground ( Figure S10c) and Rexcited ( Figure S10d) are generated with the ground-state and excited-state dielectric functions, respectively. As shown in Figure S10e, the simulated spectra agree closely with the experimental spectra (Figure 1h), suggesting that a simple exciton-induced dielectric-shift model is sufficient to explain the observed lower polariton blueshift for spatially overlapped pump/probe data. In other words, although the signal appears along the lower polariton branch dispersion, this signal reports on the exciton population rather than the EP population. Data points are black circles. The black line represents the total fit, composed from adding the yellow, blue, and red (for the exciton resonance in panel a) fit components. Note that the spectral cuts are taken at different values of k for the spatially-overlapped (-1.14 μm -1 ) vs separated (9.98 μm -1 ) pump-probe data.
To further investigate the nature of the signal in both spatially-overlapped ( Figure 1h) and spatially-separated (Figure 1i) pump-probe transient reflectance profiles, we analyze spectral line cuts at specific momenta in Figure S11. For the spatially overlapped data ( Figure S11a), line cuts along any momentum (here shown for k = -1.14 −1 ) provide an alternative visualization of the aforementioned dielectric blueshift simulated in Figure S10. The observed transient features are fit with two Lorentzian functions, where the positive component represents 'bleaching' of the lower polariton branch (disappearance of the lower polariton branch from its ground state energy, resulting in more reflectance), and the blueshifted negative component represents the new spectral location of the branch. The extra bump at 2.16 eV is from the exciton reservoir.
For spatially separated transient reflectance ( Figure S11b), with pump probe separation distances exceeding 1 micron and at a time delay of 1 ps, the transient angle-resolved spectra must report on the EP population since excitons are immobile on this timescale. Here, the transient profiles look different: the most noticeable feature is a broadening of the polariton band. The net result is a bleach of the original polariton band (fit with a positive Lorentzian), signifying that probe photons with energies and momenta matching the ground state LP are not able to enter the cavity, and new photon pass-bands (fit with a negative Gaussian function) either side of the original polariton band, indicating that photons with slightly higher and lower energies are now able to enter the cavity and populate EPs. This behavior closely resembles a blockade-like effect 21 : when pump-populated EPs are present at a specific location in space, probe photons are unable to populate EPs at the exact same energy and location. Such repulsive EP-EP interactions are known to lead to spectral broadening (imaginary self-energy) 22 , the main feature observed in Figure S11b.
These transient angle-resolved reflectance profiles thus indicate that the mechanism for contrast generation in MUPI arises from blockade-like EP-EP interactions, whereby the imaged probe field is modified by the presence of propagating pump-generated EPs: the latter change the probability for probe photons to enter the cavity at the center probe energies, resulting in enhanced or suppressed reflectivity depending on the exact probe energy and momentum selected.

Model Hamiltonian
We consider a simple one-dimensional model 23,24 with both Peierls and Holstein-type excitonphonon interactions coupled to a set of radiation modes inside an optical cavity 25 . The light-matter Hamiltonian in atomic units (ℏ = 1) is given as [25][26][27] , In the light-matter Hamiltonian expressed above, ̂ † and ̂ are the excitonic creation and annihilation operators at site = 1, 2 … located at = ⋅ with transition dipole ̂= ̂(̂ + ̂ †), ̂ † and ̂ are the photonic creation and annihilation operators of cavity mode with a frequency and polarization direction ̂, 0 is the exciton site energy, {̂} and {̂} are the position and momentum operators of a set of phonon modes with frequency p = √ / and , and characterizes local (Holstein) and non-local (Peierls) exciton-phonon coupling strengths, is the exciton-photon coupling strength and is a hopping parameter. As in past work on halide perovskites [28][29][30] , we treat the lattice classically. While this model can be studied on a two-dimensional lattice, we do not expect qualitative differences from the one-dimensional case we treat here.
For a two-dimensional cavity 26 with a confinement in the direction with a length , the transverse primary photon mode wavevector is = / . Here the refractive index of the medium is assumed as 1 for simplicity. Assuming periodic boundary conditions in the direction, such that +1 ≡ 1 , the cavity mode wavevectors in the direction become We emphasize that the exciton-phonon coupling only appears in the {| , 0⟩} subspace. This is why higher photonic character in a polariton state leads to lower effective polariton-phonon coupling. Below we have tabulated the parameters used in this work which are adapted from Ref 24,31 and not to be taken as necessarily realistic for perovskite materials. To simulate multiple photonic bands coupled to excitonic system at various detuning, we construct two different model each of which considers one exciton band and one photonic band (as was done in Fig.S4) at various detuning Δ = 0 − 0 (with 0 = 1.57 eV). The parameters that describe these two models (labelled as I and II) are provided below, Here, ≡ is used to label wavevectors instead of for simplicity. The upper and lower polariton bands are then given as ( + ) ± √( − ) 2 + 4 2 cos 2 / 0 .
For the parameters used in Model I and II, − ( ) roughly corresponds to the LP2 and LP3 in Fig.S4, respectively. We also emphasize that while for 2D materials in a cavity one must use an (N+1)⨉(N+1) Hamiltonian (where N is the number of cavity modes), it is more appropriate to use the above 2⨉2 Hamiltonian (1 excitonic band couples to 1 photon band) for each photonic band for a 3D material that extends along the cavity quantization direction. This is because for a 3D material, there are N degenerate exciton bands along kx (instead of 1 band for a single layer) where each of them only couples to one cavity mode with matching symmetry. That said, both models will provide the same physics showing ballistic or diffusive transport of polaritons depending on the exciton content.

Quantum Dynamics Approach
In this work we employ a mixed quantum-classical approach to propagate the quantum dynamics of the light-matter system. The phonon degrees of freedom (DOF) are treated classically while the electron-photon subsystem is treated quantum mechanically. The coupled motion of electronic and nuclear DOF are evolved using the mean-field Ehrenfest method.
In this work we have used a nuclear time-step of Δ = 10 fs to obtain converged dynamics. Here, we also assume that the initial excitation is localized at the center, which is achieved by shifting site indexes for each trajectory such that ⟨Ψ(0)|̂|Ψ(0)⟩ = /2. For computing the purity of the density matrix (next section), one should also pay special attention to the relative phases between the initial state prepared in different trajectories.
The phonon DOFs are sampled from a classical Boltzmann distribution, where = 1/ (T = 300K). In this work we have used 250 trajectories to converge all results.

Computing polariton coherence through the purity of the density matrix
To understand how phonons lead to decoherence of polaritonic wavepackets, and to confirm that ballistic transport is associated with polariton coherence, we compute the time-dependent purity of the reduced density matrix ̂( ) = |Ψ( )⟩⟨Ψ( )| of the exciton-photon subsystem. The purity is defined as the trace of the density matrix squared, [̂2( )]. Note that the initial localized wavepacket prepared depends on the phonon coordinates {qn} which assume different values for different trajectories; therefore, the purity at t = 0 is less than 1. We also disregard trajectories that have less than 50% overlap to some reference trajectory. Note that for low exciton content (<25% excitonic) the initial purity is very close to 1. To compare the decay of the purity at various excitation energies, we present the normalized purity [̂2( )]/ [̂2(0)], shown in Figure S13 for different excitonic fractions of the polariton. The normalized purity is 1 for a highly coherent system, and decays to 0 in the limit of an incoherent system. As expected, these results show that the decay of the purity increases with increasing exciton content. For low exciton content, such as EPs I (6.5% excitonic), II (9.0%), and III (15%), the decay of the purity is negligible over the EP lifetime. For higher exciton content (IV 25%), some decay of purity occurs on timescale of several hundreds of fs, indicating coherence is still maintained for much of the polariton lifetime. At 47% exciton content (V), the purity decays to 0.1 within 300 fs; correspondingly, we do not observe a ballistic wavefront in our simulations for 47% exciton content. These calculations confirm that the observation of ballistic propagation is directly associated with coherence of the polariton subsystem.

Numerical Details
To extract the velocities of wavefronts from our simulations, we compute the position of the wavefront ̃( ) = ⋅ ′ where ′ is determined from is obtained by solving from the expression     (d-f) Line cuts along for three different energies showing clearly-resolved interference fringes when the double slit is inserted. The fringes are not visible for the highest-energy branch, suggesting lower spatial coherence for high-exciton EPs. Note that linewidth broadening at high exciton content also contributes to lower fringe visibility. Although these double-slit measurements confirm that EPs exhibit spatial coherence in the system, they do not report on any dynamic (e.g. scattering) effects that contribute to incoherent propagation.  Figure 3f). The initial apparently immobile signal is assigned to EP-EP scattering following resonant excitation, which creates a dense population of EPs within the pump pulse temporal width of ~60 fs. In Figure 3f of the main text, we show Monte-Carlo simulations that reproduce this behavior. Figure S20. Data fits for the resonant excitation EP propagation data presented in Figure  S19.

EP propagation at room
(a-f) Temporal evolution series for EP energies of 1.94 eV, 1.91 eV, 1.77 eV, 1.70 eV, 1.65 eV and 1.59 eV, respectively. Note that for resonant excitation, a clear wavepacket-like feature is observed because there are no reservoir exciton states refilling the LP branch and causing the elongated features observed in non-resonant excitation. As such, fitting for resonant excitation experiments is performed by tracking the position of the maximum amplitude of the wavepacket, as illustrated in the figure. Figure S21. Optical images (a,d), atomic force microscopy images (b, e) and corresponding line cuts (c, f) for the layered halide perovskite and WSe2 slabs used for Figures 4b and 4c. The layered perovskite slab is 1.13 μm thick. The WSe2 slab is 69 nm thick.